Abstract
Water-hammer control strategies constitute an essential and critical task for both hydraulic designers and manufacturers to ensure the global economic efficiency and safety operations of hydraulic utilities. The primary objective of this paper is to present an alternative strategy to control water-hammer up- and down-surges, induced into a steel piping system. The proposed technique is based on replacing a short-section of the transient sensitive regions of the existing piping system by another one made of polymeric material. Two types of polymeric materials, used for the short-section and including high- or low-density polyethylene (HDPE) or (LDPE), are addressed in this study. The 1-D pressurized-pipe flow model is used to describe the hydraulic system, along with the Ramos formulation, based on two decay coefficients being used for considering the pipe-wall viscoelastic behavior and unsteady friction effects. Numerical computations were performed using the fixed-grid method of characteristics. The efficiency of the numerical model is first verified against experimental data available from the literature. Thereafter, critical flow scenarios relating to water-hammer up- and down-surges, including a cavitating flow, are revealed and discussed to point out the efficiency of the used protection technique. From the case studied, it is found that such a technique could mitigate critical water-hammer surges and, hence, might greatly enhance the reliability of the industrial hydraulic systems and urban water utilities, while safeguarding operators. Despite the available protection measures, the utilized technique can substantially soften both up- and down-surge waves induced by severe water-hammer events. It is also found that the amortization of pressure rise and pressure drop is slightly more important for the case of a short-section made of LDPE polymeric material than that using an HDPE polymeric material. It is also observed that other factors contributing to the damping rate depended upon the short-section length and diameter. In fact, the examination of the pressure peak magnitude sensitivity, with the short-section length and diameter being the controlling variables, provides optimum values of these parameters for sizing the replaced polymeric short-section.
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Abbreviations
- A :
-
Cross-sectional area of the pipe (m2)
- a 0 :
-
Elastic wave speed (m/s)
- \({{\tilde{a}}_0 }\) :
-
Adjusted elastic wave speed (m/s)
- Cr :
-
Courant number (–)
- D :
-
Main-pipe inner diameter (m)
- d short-section :
-
Diameter of the polymeric short-section (m)
- E 0 :
-
Dynamic modulus (Young’s modulus) of pipe elasticity (Pa)
- e :
-
Pipe-wall thickness (m)
- f :
-
Darcy–Weisbach friction factor (–)
- g :
-
Gravity acceleration (m/s2)
- h f :
-
Head loss per unit length (–)
- \({H=p/\gamma +z }\) :
-
Piezometric-head (m)
- J 0 :
-
Instantaneous or elastic creep-compliance (Pa−1)
- K :
-
Bulk modulus of elasticity of the fluid (Pa)
- k r1, k r2 :
-
Ramos’s unsteady decay coefficients (–)
- L :
-
Length of the main-piping system (m)
- l main-pipe :
-
Length of the modified steel pipe (m)
- l short-section :
-
Length of the polymeric short-section (m)
- \({p^{\ast} }\) :
-
Absolute pressure (Pa)
- \({p_{\rm sat} }\) :
-
Liquid saturation pressure (Pa)
- Q :
-
Flow rate (m3/s)
- Q d :
-
Average flow at downstream the air cavity during \({2\Delta t}\) period (m3/s)
- \({R=\frac{f}{2gD} }\) :
-
Pipeline resistance coefficient (–)
- t :
-
Time (s)
- x :
-
Coordinate along the pipe axis (m)
- z :
-
Elevation or pipe axis elevation (m)
- \({\alpha }\) :
-
Dimensionless parameter (–)
- \({\alpha _0 }\) :
-
Pressure-dependent volumetric ratio of gas in mixture (void fraction) (–)
- \({\Delta t }\) :
-
Time-step increment (s)
- \({\Delta x }\) :
-
Space-step increment (m)
- \({\rho }\) :
-
Fluid density (kg/m3)
- \({{\nu }'}\) :
-
Kinematic fluid viscosity (m2/s)
- \({\psi }\) :
-
Numerical weighting factor (–)
- \({\theta }\) :
-
Relaxation coefficient for the local acceleration numerical scheme (–)
- \({\forall }\) :
-
Volume of cavity (m3)
- 0:
-
Steady state
- i :
-
Section index
- ns :
-
Number of sections
- g :
-
Gas
- j :
-
Pipe index
- np :
-
Number of pipes
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Triki, A. Water-hammer control in pressurized-pipe flow using an in-line polymeric short-section. Acta Mech 227, 777–793 (2016). https://doi.org/10.1007/s00707-015-1493-1
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DOI: https://doi.org/10.1007/s00707-015-1493-1